A solid sphere of radius R and density rho is | vedclass

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(C) Let $V = \frac{4}{3} \pi R^3$ be the volume of each sphere.For the upper sphere (density $\rho$): The forces acting are gravity ($mg = V\rho g$ do

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$(A, D)$

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(C) Let $V = \frac{4}{3} \pi R^3$ be the volume of each sphere.For the upper sphere (density $\rho$): The forces acting are gravity ($mg = V\rho g$ downwards),spring force ($kx$ upwards,as it is being pulled down by the lower sphere),and buoyant force ($F_{B1} = V(2\rho)g$ upwards). At equilibrium: $V\rho g + kx = V(2\rho)g \implies kx = V\rho g = \frac{4}{3} \pi R^3 \rho g$. Thus,$x = \frac{4 \pi R^3 \rho g}{3 k}$. This confirms statement $(A)$ is correct.For the lower sphere (density $3\rho$): The forces are gravity ($mg = V(3\rho)g$ downwards),spring force ($kx$ downwards),and buoyant force ($F_{B2} = V(2\rho)g$ upwards). At equilibrium: $V(3\rho)g = V(2\rho)g + kx \implies kx = V\rho g = \frac{4}{3} \pi R^3 \rho g$. This confirms the elongation is indeed $\frac{4 \pi R^3 \rho g}{3 k}$.Since the buoyant force on the upper sphere $(2V\rho g)$ is greater than its weight $(V\rho g)$,it must be fully submerged to maintain equilibrium,as the spring provides the necessary downward force to balance the net upward buoyant force. Thus,the light sphere is completely submerged. Statement $(D)$ is correct.

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